$H \rightarrow b\overline{b}j$ at Next-to-Next-to-Leading Order Accuracy

We present the calculation of the decay $H \rightarrow b\overline{b}j$ at next-to-next-to-leading order (NNLO) accuracy. We consider contributions in which the Higgs boson couples directly to bottom quarks, i.e. our predictions are accurate to order $\mathcal{O}(\alpha_s^3 y_b^2)$ . We calculate the various components needed to construct the NNLO contribution, including an independent calculation of the two-loop amplitudes. We compare our results for the two-loop amplitudes to an existing calculation finding agreement. We present multiple checks on our two-loop expression using the known infrared factorization properties as the emitted gluon becomes soft or collinear. We use our results to construct a Monte Carlo implementation of $H \rightarrow b\overline{b}j$ and present jet rates and differential distributions in the Higgs rest frame using the Durham jet algorithm.